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Quantum Black Holes and Gauge/Gravity Duality

Badis Ydri

TL;DR

The work surveys how AdS2/ JT gravity with conformal matter, combined with the island conjecture and replica-wormhole techniques, yields a unitary Page curve for black hole evaporation. By computing quantum extremal surfaces and incorporating the island, it demonstrates how entanglement wedges extend behind horizons and reconcile the fine-grained von Neumann entropy with the Bekenstein–Hawking entropy. The replica-trick framework shows that replica wormholes dominate the Euclidean path integral, giving rise to the island rule and a consistent entropy evolution for radiation. Collectively, these results reinforce the holographic picture of information preservation and illuminate the role of higher-dimensional duals (e.g., AdS3) in a coherent island/ER=EPR story.

Abstract

From a conceptual point of view, this chapter may be viewed as an exercise in combining quantum field theory and general relativity in a controlled setting. Despite its apparent simplicity, this exercise is deeply rooted in highly non-trivial developments in superstring theory and holography, and it addresses what is arguably one of the most profound questions in quantum gravity: the fate of information in black hole evaporation. For this reason, one is naturally compelled to examine this exercise as carefully as possible, closely following the original authors. This chapter presents thus a detailed overview of recent work by Almheiri et al. and Penington on the AdS2 black hole information loss paradox and its proposed resolution within the framework of the AdS/CFT correspondence. The roles of generalized entanglement entropy, quantum extremal surfaces, the island conjecture, holography, and replica wormholes in this resolution are discussed in detail. The distinction between the von Neumann entropy and the Bekenstein-Hawking entropy in black hole physics is carefully clarified. It is shown that a phase transition at the Page time, between the trivial quantum extremal surface at the horizon and a non-vanishing quantum extremal surface located behind the horizon, leads to the correct Page curve. A simplified version of the information loss problem for the eternal AdS2 black hole, together with its resolution along similar lines, is also presented. The replica trick and its crucial role in computing entanglement entropies for various intervals in AdS2 are discussed at some length. However, the use of the replica trick in constructing replica wormholes that dominate the Euclidean path integral, thereby leading to the island rule and the correct quantum extremal surfaces, is only outlined.

Quantum Black Holes and Gauge/Gravity Duality

TL;DR

The work surveys how AdS2/ JT gravity with conformal matter, combined with the island conjecture and replica-wormhole techniques, yields a unitary Page curve for black hole evaporation. By computing quantum extremal surfaces and incorporating the island, it demonstrates how entanglement wedges extend behind horizons and reconcile the fine-grained von Neumann entropy with the Bekenstein–Hawking entropy. The replica-trick framework shows that replica wormholes dominate the Euclidean path integral, giving rise to the island rule and a consistent entropy evolution for radiation. Collectively, these results reinforce the holographic picture of information preservation and illuminate the role of higher-dimensional duals (e.g., AdS3) in a coherent island/ER=EPR story.

Abstract

From a conceptual point of view, this chapter may be viewed as an exercise in combining quantum field theory and general relativity in a controlled setting. Despite its apparent simplicity, this exercise is deeply rooted in highly non-trivial developments in superstring theory and holography, and it addresses what is arguably one of the most profound questions in quantum gravity: the fate of information in black hole evaporation. For this reason, one is naturally compelled to examine this exercise as carefully as possible, closely following the original authors. This chapter presents thus a detailed overview of recent work by Almheiri et al. and Penington on the AdS2 black hole information loss paradox and its proposed resolution within the framework of the AdS/CFT correspondence. The roles of generalized entanglement entropy, quantum extremal surfaces, the island conjecture, holography, and replica wormholes in this resolution are discussed in detail. The distinction between the von Neumann entropy and the Bekenstein-Hawking entropy in black hole physics is carefully clarified. It is shown that a phase transition at the Page time, between the trivial quantum extremal surface at the horizon and a non-vanishing quantum extremal surface located behind the horizon, leads to the correct Page curve. A simplified version of the information loss problem for the eternal AdS2 black hole, together with its resolution along similar lines, is also presented. The replica trick and its crucial role in computing entanglement entropies for various intervals in AdS2 are discussed at some length. However, the use of the replica trick in constructing replica wormholes that dominate the Euclidean path integral, thereby leading to the island rule and the correct quantum extremal surfaces, is only outlined.

Paper Structure

This paper contains 25 sections, 166 equations, 14 figures.

Table of Contents

  1. Outline
  2. The JT-CFT$_2$ system: A first look
  3. Eternal ${\bf AdS}^2$ black hole
  4. ${\bf AdS}^2$ Black hole formed from gravitational collapse
  5. Quantum correction and coupling to a heat bath
  6. More on the boundary theory and the Schwarzian
  7. More on JT gravity coupled to conformal matter
  8. The eternal ${\bf AdS}^2$ black hole
  9. The coupling to conformal matter
  10. The Hartle-Hawking state
  11. An evaporating ${\bf AdS}^2$ black hole
  12. Entropy and quantum extremal surface in JT gravity
  13. Entropies, Ryu-Takayanagi formula and the island conjecture
  14. Entanglement entropy of an interval in CFT$_2$ and ${\bf AdS}^2$
  15. The late time quantum extremal surfaces
  16. ...and 10 more sections

Figures (14)

  • Figure 2: The whole ${\bf AdS}^2$ theory maps to the point $\sigma=0$ where the boundary is located and where the dual quantum mechanics lives (holographic correspondence).
  • Figure 3: The vanishing surface (before Page time) and quantum extremal surface or QES (after Page time).
  • Figure 4: Entanglement wedges before (first diagram) and after (second diagram) the Page time. The radiation wedge is drawn in yellow whereas the black hole wedge is drawn in purple. The island is counted in the radiation wedge after the Page time.
  • Figure 5: The RS model.
  • Figure 6: Three equivalent description of ${\bf AdS}^2$ black hole.
  • ...and 9 more figures